See also:

• ARITHMETIC (Gr. apeOµ7run7, sc. TEXVn, the art of counting, from (ipLBµos, number)

The various stages in the study of arithmetic may be arranged in different ways, and the arrangement adopted must be influenced by the purpose in view. There are three See also:

• MAIN (from the Aryan root which appears in " may " and " might," and Lat. magnus, great)
• MAIN (Lat. Moenus)

All these aspects are of importance to the teacher: the logical, in See also:

• ORDER
• ORDER (through Fr. ordre, for earlier ordene, from Lat. ordo, ordinis, rank, service, arrangement; the ultimate source is generally taken to be the root seen in Lat. oriri, rise, arise, begin; cf. " origin ")
• ORDER, HOLY

6. The See also:

• PRESENT

. . are used sometimes as ordinals, i.e. to denote the See also:

• PLACE (through Fr. from Lat. platea, street; Gr. IrAar6s, wide)

1876. It might appear as if the See also:

• WRITING
• WRITING (the verbal noun of " to write," O. Eng. writan, to inscribe)

Hence, if there are 365 days in a year, the See also:

• MOTH

(ii) Method of Counting.—The second method consists in taking a series of names or symbols for the first few numbers, and then repeating these according to a See also:

• REGULAR

. . , or symbols such as I, 2, 3, . . ; this sequence being always the same. Ifwe take a set of concrete objects, and name them in See also:

• SUCCESSION (Lat. successio, from succedere, to follow after)

, for instance, are formed according to a definite law; and in giving 253 as the number of a set of objects we mean that if we attach to them the symbols I, 2, 3, ... in succession, according to this law, the symbol attached to the last object will be 253. If we say that this act of attaching a symbol has been performed 253 times, then 253 is an abstract (or pure) number. Underlying this definition is a certain See also:

• ASSUMPTION, FEAST OF

The £I is termed the unit, A numerical quantity, therefore, represents a certain unit, taken a certain number of times. If we take £3 twice, we get £6; and if we take 3s. twice, we get 6s., i.e. 6 times Is. Thus arithmetical processes deal with numerical quantities by dealing with numbers, provided the unit is the same throughout. If we retain the unit, the arithmetic is, concrete; if we ignore it, the arithmetic is abstract. But in the latter case it must always be understood that there is some unit concerned, and the results have no meaning until the unit is reintroduced. II. NOTATION, NUMERATION AND NUMBER-IDEATION 12. Terms used.—The See also:

• REPRESENTATION

Notation of Numbers.—The system which is now almost universally in use amongst civilized nations for representing cardinal numbers is the See also:

• HINDU

. We then have, e.g., 240 to denote two hundreds and four tens; and we may now adopt a See also:

• UNIFORM

(i) If the numbers were written down in succession, they would naturally proceed from See also:

• LEFT

(iii) In making out lists, schedules, mathematical tables (e.g. a multiplication-table), statistical tables, &c., the numbers are written vertically downwards. In the case of lists and schedules the numbers are only ordinals; but in the case of mathematical or statistical tables they are usually regarded as cardinals,though, when they represent values of a continuous quantity, they must be regarded as ordinals (§§ 26, 93). (iv) In graphic representation measurements are usually made upwards; the See also:

• ADOPTION (Lat. adoptio, for adoptalio, from adoptare, to choose for oneself)

Thus 1878 is MDCCCLXXVIII, i.e. thousand five-hundred hundred hundred hundred fifty ten ten five one one one. The system is therefore essentially a cardinal and grouping one, i.e. it represents a number as the sum of sets of other numbers. It is therefore remarkable that it should now only be used for ordinal purposes, while the Hindu system, which is ordinal in its nature, since a single series is constantly repeated, is used almost exclusively for cardinal numbers. This fact seems to illustrate the truth that the counting principle is the fundamental one, to which the See also:

• INTERPRETATION (from Lat. interpretari, to expound, explain, inter pres, an agent, go-between, interpreter; inter, between, and the root pret-, possibly connected with that seen either in Greek 4 p4'ew, to speak, or irpa-rrecv, to do)

The Roman system, except for the use of symbols for five, fifty, &c., is also in the denary scale, though expressed in a different way. The introduction of these other symbols produces a See also:

• COMPOUND
• COMPOUND (from Lat. componere, to combine or put together)

There are, therefore, four denominations, the bases for See also:

• CONVERSION (Lat. conversio, from convertere, to turn or ,change)

A quantity expressed in two or more denominations is usually called a compound number or compound quantity. The former term is obviously incorrect, since a quantity is not a number; and the latter is not very suggestive. For agreement with the terminology of fractional numbers (§ 62) we shall describe such a quantity as a mixed quantity. The letters or symbols descriptive of each denomination are usually placed after or (in actual calculations) above the figures denoting the numbers of the corresponding units; but in a few cases, e.g. in the case of £, the symbol is placed before the- figures. There would be great convenience in a general adoption of this latter method; the combination of the two methods in such an expression as £I23, 16s. 4zd. is especially awkward. 18. Numeration.—The names of numbers are almost wholly based on the denary scale; thus eighteen means eight and ten, and twenty-four means twice ten and four. The words eleven and twelve have been supposed to suggest etymologically a denary basis (see, however, NUMERAL). Two exceptions, however, may be noted. (i) The use of dozen, See also:

• GROSS

(ii) The See also:

• SCORE (O.E. scor, from sceran, to cut, notch, cf. " shear ")

(iv) A See also:

• COMPROMISE (pronounced compromize; through Fr. from Lat. compromittere)

As regards the first of these two-examples, however, it would be more correct to write 1,82o for the former of the two meanings (cf. § 13). (ii) The symbols II and 12 are read as eleven and twelve, not (except in elementary teaching) as ten-one and ten-two. (iii) The names of the numbers next following these, up to 19 inclusive, only faintly suggest a ten. This difficulty is not always recognized by teachers, who forget that they themselves had to be told that eighteen -means eight-and-ten. - (iv) Even beyond twenty, up to a hundred, the word ten is not used in numeration, e.g. we say See also:

• THIRTY

This is the case, for instance, in the See also:

• CELTIC

Later, a system similar to the See also:

• HEBREW

Again, in the use of decimals, it is unusual to give less than two figures. Thus 3.142 or 3.14 would be quite intelligible; but 3.1 does not convey such a See also:

• GOOD, JOHN MASON (1764-1827)

The limit for most adults, under favourable conditions, is about four. Under certain conditions it is less; thus IIII, the old Roman notation for four, is difficult to distinguish from III, and this may have been the main See also:

• REASON (Lat. ratio, through French raison)

The proportion of persons in whom number-forms exist has been variously estimated; but there is reason to believe that the forms arise at a very See also:

• EARLY
• EARLY, JUBAL ANDERSON (1816-1894)

Probably the latter is almost universally the case. These forms are See also:

• DEVELOPED

Thus many of the Australasian and See also:

• SOUTH
• SOUTH, ROBERT (1634–1716)

It has been suggested that names of this kind may have been the origin of the numeral words of different races; but it is improbable that direct visual perception would See also:

• LEAD
• LEAD (pronounced iced)

Representation of Geometrical Magnitude by Number.—The application of arithmetical methods to geometrical measurement presents some difficulty. In reality there is a transition from a cardinal to an ordinal system, but to an ordinal system which does not agree with the original ordinal system from which the cardinal system was derived. To see this, we may represent ordinal numbers by the ordinary numerals 1, 2, 3, . . and cardinal numbers by the Roman I, II, III, . . . Then in the earliest stage each object counted is indivisible; either we are counting it as a whole, or we are not counting it at all. The symbols 1, 2, 3, . . . then refer to the individual objects, as in fig. 1; this is the primary See also:

• FIGS

3 shows how the name II of the whole is determined by the name 2 of the last one counted. When now we pass to geometrical measurement, each " one " is a thing which is itself divisible, and it cannot be said that at any moment we are counting it; it is only when one is completed that we can count it. The names 1, 2, 3, ... for the individual objects cease to have an intelligible meaning, and measurement is effected by the cardinal numbers I, II, III, . . . , as in fig. 4. These cardinal numbers have now, however, come to denote individual points in the line of measurement, i.e. the points of separation of the individual units of length. The point III in fig. 4 does not include the point II in the same way that the number III includes the number II in fig; 2, and the points must. therefore be denoted by the ordinal numbers 1, 2, 3, . . as in fig. 5, the zero o falling into its natural place immediately before the commencement of the first unit. Thus, while arithmetical numbering refers to units, geometrical numbering does not refer to units but to the intervals between units.

(i.) Preliminary 27. Equality and Identity. =There is a certain difference between the use of words referring to equality and identity in I 2 3 I 2 3 • I • II • III FIG. 3. I 2 3 III 0 1 2 FIG. 5. arithmetic and in algebra respectively; what' is an equality in the former becoming an identity in the latter. Thus the statement that 4 times 3 is equal to 3 times 4, or, in abbreviated form, 4X3 =3 X4(§ 28), is a statement not of identity but of equality; i.e. 4 X 3 and 3 X4 mean different things, but the operations which they denote produce the same result. But in algebra a X b= b X a is called an identity, in the sense that it is true whatever a and b may be; while nXX=A is called an See also:

• EQUATION (from Lat. aequatio, aequare, to equalize)

Symbols of Operation.— The failure to observe the distinction between an identity and an equality often leads to loose reasoning; and in order to prevent this it is important that definite meanings should be attached to all symbols of operation, and especially to those which represent elementary operations. The symbols — and = mean respectively that the first quantity mentioned is to be reduced or divided by the second; but there is some vagueness about + and X. In the present article a+b will mean that a is taken first, and b added to it; but a X b will mean that b is taken first, and is then multiplied by a. In the case of numbers the X may be replaced by a dot; thus 4.3 means 4 times 3. When it is necessary to write the multiplicand before the multiplier, the symbol >e will be used, so that bse a will mean the same as a X b. 29. Axioms.—There are certain statements that are some-times regarded as axiomatic; e.g. that if equals are added to equals the results are equal, or that if A is greater than B then A+X is greater than B+X. Such statements, however, are capableof logical See also:

• PROOF (in M. Eng. preove, proeve, preve, &°c., from O. Fr . prueve, proeve, &c., mod. preuve, Late. Lat. proba, probate, to prove, to test the goodness of anything, probus, good)

Except with very small numbers, addition and subtraction, on the grouping system, involve analysis and rearrangement. Thus the sum of 8 and 7 cannot be expressed as ones; we can either form the whole, and regroup it as ro and 5, or we can split up the 7 into 2 and 5, and add the 2 to the 8 to form ro, thus getting 8+7 =8+ (2+5) = (8+2) +5=10+5= 15. For larger numbers the rearrangement is more extensive; thus 24+31= (20+4) + (30+ I) = (20+30) + (4+ I) = 50+5 = 55, the processbeing still more complicated when the ones together make more than ten. Similarly we cannot subtract 8 from 15, if 15 means 1 ten + 5 ones; we must either write 15—8=(I0+5)—8= (ro—8)+5=2+5=7, or else resolve the 15 into an inexpressible number of ones, and then subtract 8 of them, leaving 7. Numerical quantities, to be added or subtracted, must be in the same denomination; we cannot, for instance, add 55 shillings and roo pence, any more than we can add 3 yards and 2 metres. 31. Relative Position in the Series.—The above method of dealing with addition and subtraction is synthetic, and is appropriate to the grouping method of dealing with number. We commence with processes, and see what they lead to; and thus get an idea of sums and differences. If we adopted the counting method, we should proceed in a different way, our method being See also:

• ANALYTIC (the adjective of " analysis," q.v.)

The series are shown below, the numbers being placed horizontally for convenience of printing, instead of vertically (§ 14): I 2 3 4 5 6 7 8 9 1 2 3 4 1 2 3 4 5This exhibits 9 as the sum of 4 and 5; it being understood that the sum of 4 and 5 means that we add 5 to 4. That this gives the same result as adding 4 to 5 may be seen by reckoning the series backwards. It is convenient to introduce the zero; thus o I 2 3 4 5 6 7 8 9 o I 2 3 4 5 indicates that after getting to 4 we make a fresh start from 4 as our zero. To subtract, we may proceed in either of two ways. The subtraction of 4 from 9 may mean either " What has to be added to 4 in order to make up a total of 9," or " To what has 4 to be added in order to make up a total of 9." For the former meaning we count forwards, till we get to 4, and then make a new count, parallel with the continuation of the old series, and see at what number we arrive when we get to 9. This corresponds to the concrete method, in which we have 9 objects, take away 4 of them, and recount the remainder. The alternative method is to retrace the steps of addition, i.e. to count backwards, treating g of one (the standard) series as corresponding with 4 of the other, and finding which number of the former corresponds with o of the latter. This is a more advanced method, which leads easily to the idea of negative quantities, if the subtraction is such that we have to go behind the o of the standard series. 32. Mixed Quantities.—The application of the above principles, and of similar principles with regard to multiplication and division, to numerical quantities expressed in any of the diverse British denominations, presents no theoretical difficulty if the successive denominations are regarded as constituting a varying scale of notation (§17). Thus the expression 2 ft. 3 in. implies that in counting inches we use o to eleven instead of o to g as our first repeating series, so that we put down i for the next denomination when we get to twelve instead of when we get to ten.

Similarly 3 yds. 2 ft. means yds. o I 2 3 ft. O I 2 0 I 2 O I 2 0 I 2 The practical difficulty, of course, is that the addition of two numbers produces different results according to the scale in which we are for the moment proceeding; thus the sum of g and 8 is 17, 15, 13 or I I according as we are dealing with shillings, pence, pounds (See also:

• AVOIRDUPOIS

.; which are written r X 5d., 2 X 5d., 3 X 5d., . .. (§ 28). This process is repetition, and the quantities 1 X 5d., 2 X 5d., 3 X 5d are the successive multiples of 5d. If, on the other hand, we have a sum of 5s., and treat a shilling as being See also:

• EQUIVALENT

£3 4f. The essential portions of these statements, from the arith- t 20 is. . 12d. metical point of view, may be exhibited in the form of the diagrams A and B A B I2 £720 £t 20S. i boy 7 apples is. I2d. £2880 £3 6os. 720d. 288of. 3 boys 21 apples 5s. 6od. or more briefly, as in C or C' and D or D': C C' D I 7 apples 3 21 apples 3 the general arrangement of the diagram being as shown in E or E' E E' Multiplication is therefore equivalent to completion of the diagram by entry of the product.

36. Multiple-Tables.—The diagram C or D of § 35 is part of a See also:

• COMPLETE

I I yds. oft. 52077 .... . 4 4 8 36 5s. 8d. 14 yds. 2 ft. 69436 • . 5 5 10 45 7s. id. 18 yds. i ft. 86795 • . It is to be considered that each column may extend downwards indefinitely. 37.

Successive Multiplication.—In multiplication by repetition the unit is itself usually a multiple of some other unit, i.e. it is a product which is taken as a new unit. When this new unit has been multiplied by a number, we can again take the product as a unit for the purpose of another multiplication; and so on indefinitely. Similarly where multiplication has arisen out of the subdivision of a unit into smaller units, we can again subdivide these smaller units. Thus we get successive multiplication; but it represents quite different operations according as it is due to repetition, in the sense of § 34, or to subdivision, and these operations will be exhibited by different diagrams. Of the two diagrams below, A exhibits the successive multiplication of £3 by 20, 12 and 4, and B the successive reduction of £3 to shillings, pence and farthings. The principle on which the diagrams are constructed is obvious from § 35. It should be noticed that in multiplying 6 by 20 we find the value of 20.3, but that in 38. Submultiples.—The relation of a unit to its successive multiples as shown in a multiple-table is expressed by saying that it is a submultiple of the multiples, the successive sub-multiples being one-See also:

• HALF

8d ; these being written " of 2S. rod.," " a of 4s. 3d.," " 4 of 5s. 8d.," The relation of submultiple is the converse of that of multiple; thus if a is i of b, then b is 5 times a. The determination of a sub-multiple is therefore equivalent to completion of the diagram E or E' of § 35 by entry of the unit, when the number of times it is taken, and the product, are given. The operation is the converse or repetition; it is usually called See also:

• PARTITION

= 12d., it follows that the number of pence in one-fourth of I2d. is equal to the quotient when 12 pence are formed into units of 4d.; each of these numbers being said to be obtained by dividing 12 by 4. The term division is therefore used in text-books to describe the two processes described in §§ 38 'and 39; the product mentioned in § 34 is the See also:

• DIVIDEND (Lat. dividendum, a thing to be divided)

12d. 21 apples 5 6od. 5 6od. I Unit Unit Number Product Number Product I Number Product Unit 12d. is. 6od. in § 37; A representing the determination of -6 of of ,-x of 288o farthings, and B the conversion of 288o farthings into £. A B (iv.) Properties of Numbers. (A) Properties not depending on the Scale of Notation. 43• Powers, Roots and Logarithms.—The standard series 1, 2, 3, .. is obtained by successive additions of x to the number last found. If instead of commencing with x and making successive additions of r we commence with any number such as 3 and make successive multiplications by 3, we get a series 3, 9, 27, . . . as shown below the line in the margin. The first memo I = 3° n° ber of the series is 3; the second is the product of I 3=3' n' two numbers, each equal to 3; the third is the See also:

• PRO

3 27=3' n4 These are written 31 (or 3), 32, :3', 34, . . where 4 SI -3 n nP denotes the product of p numbers, each equal to : n. If we write nP = x, then, if any two of the three numbers n, p, u are known, the third is determinate. If we know n and p, p is called the index, and n, n2, . .. nP are called the first power, second power, . . . pth power of n, the series itself being called the power-series. The second power and third power are usually called the square and cube respectively. If we know p and N, is is called the pth See also:

• ROOT (late O.E. rot, adopted from Scand., cf. Norw. and Swed. rot, Dan. rod; the true O.E. word was wyrt, plant, represented in Ger. Wurz or Wurzel; the ultimate root is the same in both words, and is seen in Lat. radix)
• ROOT, ELIHU (1845– )

The calculation of a logarithm can be performed by successive divisions; evolution requires special methods. The above definitions of logarithms, &c., relate to cases in which n and p are whole numbers, and are generalized later. 44• Law of Indices.—If we multiply nP by 19, we multiply the product of p n's by the product of q n's, and the result is therefore nP-l-4. Similarly, if we divide nP by n4, where q is less than p, the result is nP-Q. Thus multiplication and division in the power-series correspond to addition and subtraction in the index-series, and See also:

• VICE

If 7 8 8 any number N occurs in the 8 vertical series commencing 9 9 with a number is (other than Io IO 12 12 IO I) then n is said to be a See also:

• FACTOR (from Lat. facere, to make or do)

These are called prime factors. The following are the most important properties of numbers in reference to factors: (i) If a number is a factor of another number, it is a factor of any multiple of that number. (ii) If a number is a factor of two numbers, it is a factor of their sum or (if they are unequal) of their difference. (The words in brackets are inserted to avoid the difficulty, at this stage, of saying that every number is a factor of o, though it is of course true that o. n=o, whatever is may be.) (iii) A number can be resolved into prime factors in one way only, no account being taken of their relative order. Thus 12=2X2X3=2X3X2=3X2X2, but this is regarded as one way only. If any prime occurs more than once, it is usual to write the number of times of occurrence as an index; thus 144=2X2X2X2X3X3=24 32. The number r is usually included amongst the primes; but, if this is done, the last See also:

• PARAGRAPH

If two numbers have no factor in common (except x) each is said to be prime to the other. The multiples of 2 (including 1.2) are called even numbers; other numbers are See also:

• ODD (in middle English odde, from old Norwegian oddi, an angle of a triangle; the old Norwegian oddamann is used of the third man who gives a casting vote in a dispute)

Thus, since 144 = 24. 32, and 756 = 22. 33 7, the L.C.M. of 144 and 756 is 24 33 7. It is clear, from comparison with the last paragraph, that the product of the G.C.D. and the L.C.M. of two numbers is equal to the product of the numbers themselves. This gives a rule for finding the L.C.M. of two numbers. But we cannot apply it to finding the L.C.M. of three or more number; if we cannot resolve the numbers into their prime factors, we must find the L.C.M. of the first two, then the L.C.M. of this and the next number, and so on. (B) Properties depending on the Scale of Notation. 48. Tests of Divisibility.-The following are the See also:

• PRINCIPAL

12d. Id. 3 6os. 288of. 720d. 288of. 4 I I2 I 72of. 6of. 20 I 3f. (viii) by 1 1 if the difference between the sum of the 1st, 3rd, sth, . .. digits and the sum of the 2nd, 4th, 6th, . is zero or divisible by Ir. (ix) To find whether a number is divisible by 7, 1r or 13, arrange the number in groups of three figures, beginning from the end, treat each group as a separate number,; and then find the difference between the sum of the 1st, 3rd, ... of these numbers and the sum of the 2nd, 4th, .

. Then, if this difference is zero or is divisible by 7, II or 13, the original number is also so divisible; and conversely. For example, 31521 gives 52f -31 ..490, and therefore is divisible by7,. but not by.II or 13. 49. Casting out Nines is a process based on (vi) of the last paragraph. The remainder when a number is divided by g is equal to the remainder when the sum of its digits is divided by 9. Also, if the remainders when two numbers are divided by 9 are respectively a and b, the remainder when their product is divided by 9 is the same as the remainder when a.b is divided by g. This gives a rule for testing multiplication, which is found in most text-books. It is doubtful, however, whether such a rule, giving a test which is necessarily incomplete, is of much educational value. (v.) Relative Magnitude. _ 5o. Fractions.—A fraction of a quantity is a suhmultiple, or a multiple of a submultiple, of that quantity. Thus, since 3XIs.

5d.=4s. 3d., 1s: -5d. may be denoted by i of 4S. 3d.; and any multiple of Is. sd., denoted by nXrs. 5d., may also be denoted by of 4S. 3d. We therefore use "a of A" to mean that we find a quantity X such that aXX=A, and then multiply X by n. It must be noted (i) that this is a definition of "a of, " not a definition of "a,"and (ii) that it is not necessary that t should be less than a. 51. Subdivision of Submultiple.—By r. of A we mean 5 times the unit, 7 times which is A. If we regard this unit as being 4 times a lesser unit, then A_is 7.4 times this lesser unit, and 4 of A is 5.4 times the lesser unit. Hence 4 of A is equal to 4 of A; and, conversely, 4 of A is equal to 4- of A. Similarly each of these is equal to 4 of A.

Hence the value of a fraction is not altered by substituting for the numerator and denominator the corresponding numbers in any other column of a multiple-table (§ 36). If we write 7+55_44In the form 44'.7 5 we may say that the value of a fraction is not altered by multiplying or dividing the numerator and denominator by any number. 52. Fraction of a Fraction.—To find ~-; of . 4 of A we must convert ? of A into 4 times some unit. This is done by the pre- ceding paragraph. For 4 of A=5:1 of A= 4.1 of A; i.e. it is 7.4 4 times a unit which is itself 5 times another unit, 7.4 times which is A. Hence, taking the former unit 11 times instead of 4 times, See also:

• LOFT (connected with " lift," i.e. raised in the air; O. Eng. lyft; cf. Ger. Luft; the French term is grenier and Ger. Boden)

Hence the former is greater than the latter; their sum is;$- of A; and their difference is of A. Thus the fractions must be reduced to a common denominator. This denominator must, if the fractions are in their lowest terms (§ 54), be a multiple of each of the denominators; it is usually most convenient that it should be their L.C.M. (§ 47). 54. Fraction in its Lowest Terms.—A fraction is said to be in its lowest terms when its numerator and denominator have no common 1 7d. 10 5S. See also:

• TOD, JAMES (1782-1835)

55. Diagram of Fractional Relation.—To find -T of 14s. we have to take lo of the units, 24 of which make up 14s. Hence the required amount will, in the multiple-table of § 36, be opposite 10 in the column in which the amount opposite 24 is 14s.; the quantity at the head of this column, representing the unit, will be found to be 7d. The elements of the multiple-table with which we are concerned are shown in the diagram in the margin. This diagram serves equally for the two statements that (i) 4 of 14s. is 5s. iod., (ii) -ofss. iod: is 14s. The two statements are in fact merely different aspects of a single relation, considered in the next See also:

• SECTION
• SECTION (Lat. sectio, cutting, secare, to cut)

For this reason the ratio of a to b is sometimes written but the more correct method is to write it a:b. If two quantities or numbers P and Q are to each other in the 'ratio of p to q, it is clear -from the diagram that p times Q= q times P, so that Q = p of P. 57. Proportion.—If from any two columns in the table of § 36 we remove the numbers or quantities in any two rows, we get a diagram such as that here shown. The pair of compartments on either See also:

• SIDE
• SIDE (mod. Eski Adalia)

Laws of Arithmetic.—The arithmetical processes which we have considered in reference to positive integral numbers are subject to the following laws: (i) Equalities and Inequalities.—The following are sometimes called Axioms (§ 29), but their truth should be proved, even if at an early stage it is assumed_ The symbols " > " and " < mean respectively " is greater than " and "is less than." The numbers represented by a, b, c, x and m are all supposed to be positive. A 145. S. IO . 12 5s: iod. 14S., 2S. Iod. 8s. 6d. 7 yds. I ft. 22 ,yds..

P (a) If a=b, and b=c, then a=c; (b) If a=b, then a+x=b+x, and a—x=b—x; (c) If a>b, then a+x>b+x, and a—x>b—x; (d) If a<b, then a+x<b+x, and a—x<b—x; (e) If a=b, then ma=inb, and a=m=b=m; (f) If a>b, then ma>mb, and a-m>b+m; (g) If a<b, then ma<mb, and a=m<b=m. (ii) Associative Law for Additions and Subtractions.—This law includes the rule of signs, that a—(b—c)=a—b+c; and it states that, subject to this, successive operations of addition or sub-See also:

• TRACTION (Lat. trahere, to draw)

In this way, we arrive at (i) negative numbers, (ii) fractional numbers, (iii) surds, (iv) logarithms (in the ordinary sense of the word). 6o. Simple Formulae.—The following are some simple formulae which follow from the laws stated in § 58. (i) (a+b+c+ ...) (p+q+r+ ...) = (ap+aq+ar+ ... )+ (bp+hq+br+ . . . )+(cp+cq+cr+ . . . )+ ... ; i.e. the pro-duct of two or more numbers, each of which consists of two or more parts, is the sum of the products of each part of the one with each part of the other. (ii) (a+b) (a—b)=a2—b2; i.e. the product of the sum and the difference of two numbers is equal to the difference of their squares. (iii) (a+b)2=See also:

• A2(z)

V. NEGATIVE NUMBERS 61. Negative Numbers may be regarded as resulting from the commutative law for addition and subtraction. According to this law, 10+3+6— 7= Io+3—7+6=3+6—7+I0 =&c. But, if we write the expression as 3—7+6+1o, this means that we must first subtract 7 from 3. This cannot be done; but the result of the subtraction, if it could be done, is something which, when 6 is added to it, becomes 3—7+6=3+6-7=2. The result of 3—7 is the same as that of o—4; and we may write it " -4," and See also:

• CALL (from Anglo-Saxon ceallian, a common Teutonic word, cf. Dutch kallen, to talk or chatter)

VI. FRACTIONAL AND DECIMAL NUMBERS 62. Fractional Numbers.—According to the definition in § 5o the quantity denoted by 1 of A is made up of a number, 3, and a unit, which is one-See also:

• SIXTH

. , are then fractional numbers, their relation to ordinary or integral numbers being that n times n times is equal top times. This relation is of exactly the same kind as the relation of the successive digits in numbers expressed in a scale of notation whose Ones. Sixths. base is n. Hence we can treat the fractional numbers h See also:

• AVE, or FIRN

The result of taking 13 sixths of A is, then seen to be the same as the result of taking twice A and one-sixth of A, so that we may regardas being equal to 21. A fractional number is called a proper fraction or an improper fraction according as the numerator is or is not less than the denominator; and an expression such as 21 is called a, mixed number. An improper fraction is therefore equal either to an integer or to -a mixed number. It will be seen from § 17 that ,a mixed number corresponds with what is there called a mixed quantity. Thus £3, 17s. is a mixed quantity, being ex- ' o 2 0 B Ones. Thirds. Sixths. o o o I i o 2 o I I o o thing as 7 times -7.3. Hence '4- times 4-= 4'•times 7 times -733 =5 times 7 3 = . The rule for multiplying a fractional number 7.3 by a fractional number is therefore the same as the rule for finding a fraction of a fraction. 66. Division of Fractional Numbers.-To divide 4- by S. is to find a number (i.e. a fractional number) x such that 4- times x is equal to 1.

But 4- times 7 times x is, by the last section, equal to x. Hence x is equal to s times 4. Thus to divide by a fractional number we must multiply by the number obtained by inter-changing the numerator and the denominator, i.e. by the reciprocal of the original number. If we divider by 7 we obtain, by 'this rule, g. Thus the reciprocal of a number may be defined as the number obtained by dividing i by it. This definition applies whether the original number is integral or fractional. By means of the present and the preceding sections the rule given in § 63 can be extended to the statement that a fractional number is equal to the number obtained by multiplying its numerator and its denominator by any fractional number. 67. Negative Fractional Numbers.—We can obtain negative fractional numbers in the same way that we obtain negative integral numbers ; thus — or - -A means that 7 or 4A is taken negatively. 68. See also:

• GENESIS
• GENESIS (Gr. 'ybeo-es, becoming; the term being used in English as a synonym for origin or process of coming into being)

; but, by extending the meaning of" times " as in § 62, we can say that 3 in. isr times I r in., and therefore call IN the quotient when 3 in. is divided by r r in. Hence, if p and n are numbers, p is sometimes regarded as denoting the result of dividing p by n, whether p and n are integral or fractional (mixed numbers being included in fractional). The idea and properties of a fractional number having been explained, we' may now call it, for brevity, a fraction. Thus "4 of A " no longer means two of the units, three of which make up A; it means that A is multiplied by the fraction 4-, i.e. it means the same thing as " 4 times A." 69. Percentage.—In order to deal, by way of comparison or addition or subtraction, with fractions which have different denominators, it is necessary to reduce them to a common denominator. To avoid this difficulty, in practical life, it is usual to confine our operations to fractions which have a certain standard denominator. Thus (§ 79) the See also:

• ROMANS
• ROMANS, EPISTLE TO THE

This is convenient, e.g. for expressing rates in tke pound; thus 15 % denotes the process of taking 3S. for every r, i.e. a rate of 3s. in the £. In applications to money " per cent." sometimes means "per £loo." Thus " £3, 17s. 6d. per cent." is really the complex 17,4, fraction 3 20 . - See also:

• IOO

The difference between this approximate percentage and the true value is less than %, i. e. is less than -Ts If the numerator of the fraction consists of an integer and 3 2-e.g. in the case of 8=1Oo it is uncertain whether we should take the next lowest or the next highest integer. It is best in such cases to retain the I; thus we can write a =372 %'372• 72. Addition and Subtraction of Percentages.—The sum or difference of two percentages is expressed by the sum or difference of the numbers expressing the two percentages. 73.-Percentage of a Percentage.—Since 37 % of r is expressed by 0'37, 37 % of r % (i.e. of o•oi) might similarly be expressed by 0.00.37. The second point, however, is omitted, so that we write it 0.0037 or 0037, this expression meaning 13,17 of f o 0 = 3 o~~• On the same principle, since 37 % of 45 % is equal to See also:

• POLO (Tibetan pulu, ball)
• POLO, GASPAR GIL (?153o-1591)
• POLO, MARCO (c. 1254-1324)

Thus 273105, 0 can be written 27.1530. This number, expressed in terms of the fraction i o or .See also:

• DOOR (corresponding to the Gr. Bbpa,. Lat. fores or valvae; the English word, with other forms common in allied languages, comes from the same Indo-European stem as the Gr. Obpa and Lat. fares)

It has already been explained (§ 62) that ?-times is an operation such that 4-times 7 times is equal to 5 times. Hence we must express 4-, which itself means 4- times, as being 7 times something. This is done by multiplying both numerator and denominator by 7; i.e. 4- is equal to .3' which is the same of 116, would be 271. J30, but expressed in terms of loo it would be '271530. Fractions other than decimal fractions are usually called vulgar fractions. 75. Decimal Numbers.—Instead of regarding the •1S3 in 27'153 as meaning -,)M-, we may regard the different figures in the expression as denoting numbers in the successive orders of submultiples of I on a denary scale. Thus, on the grouping system, 27.153 will mean 2.Io-f7+1/ro+5/See also:

• IO2

Sums and Differences of Decimals.—To add or subtract decimals, we must reduce them to the same denomination, i.e. if one has more figures after the decimal point than the other, we must add sufficient o's to the latter to make the numbers of figures equal. Thus, to add 5'413 to 3.8, we must write the latter as 3.800. Or we may treat the former as the sum of 5.4 and •013, and recombine the •013 with the sum of 3.8 and 5'4• 77. Product of Decimals.—To multiply two decimals exactly, we multiply them as if the point were absent, and then insert it so that the number of figures after the point in the product shall be equal to the sum of the numbers of figures after the points in the original decimals. In actual practice, however, decimals only represent approximations, and the process has to be modified (§ Iu). 78. Division by Decimal.—To divide one decimal by another, we must reduce them to the same denomination, as explained in § 76, and then omit the decimal points. Thus 5'413+3'8= 6-?In=5413+3800. 7g. Historical Development of Fractions and Decimals.—The fractions used in See also:

• ANCIENT
• ANCIENT (also spelt ANTIENT; derived, through the Fr. ancien, old, from the late Lat. antianum, from ante, before)

Except in the case of i and a, the fraction was expressed by the denominator, with a special symbol above it. The Babylonians expressed numbers less than r 'by the numerator of a fraction with denominator 6o; the numerator only being written. The choice of 6o appears to have been connected with the reckoning of the year as 36o days; it is perpetuated in the present subdivision of angles. The Greeks originally used unit-fractions, like the Egyptians; later they introduced the sexagesimal fractions of the Babylonians, extending the system to four or more successive subdivisions of the unit representing a degree. They also, but apparently still later and only occasionally, used fractions of the modern kind. In the sexagesimal system the numerators of the successive fractions (the denominators of which were the successive powers of 6o) were followed by' , , ", "", the denominator not being written. This notation survives in reference to the See also:

• MINUTE
• MINUTE (Lat. minutes, small; minuere, to make less)

Various systems were tried before the present,notation came to be generally accepted. Under one system, for instance, the continued sum 5 7 X 5X 5-f-8 X 7 X 5 would be denoted by 8 5; this is somewhat similar in principle to a decimal notation, but with digits taken in the See also:

• REVERSE

These may be divided into three classes, as follows: (i) The fraction of a concrete quantity may itself not exist as a concrete quantity, but be represented by a token. Thus, if we take a shilling as a unit, we may divide it into 12 or 48 smaller units; but corresponding coins are not really portions of a shilling, but objects which help us in counting. Similarly we may take the See also:

• FARTHING (A.S. Jeortha, fourth, ring, diminutive)

In weighing an object with ounce-weights the fact that it weighs more than I lb 3 oz. but less than r lb 4 oz. does not of itself suggest the See also:

• NECESSITY (Lat. necessitas)

§ 71) that the number differs from the true value by less than one-half of the unit represented by 1 in the last place of decimals. For instance, • 143 represents + correct to 3 places of decimals, since it differs from it by less than •o005. The final figure, in a case like this, is said to be corrected. This method is not good for See also:

• COMPARATIVE

It should be observed that the numerical value of the error is to be subtracted from or added to the stated value according as the error is positive or negative: (iii) The limit of error can be expressed as a fraction of the number as stated. Thus += '143= *0005 can be written += 143(1=a-)• 83. Accuracy after Arithmetical Operations.—If the numbers which are the subject of operations are not all exact, the accuracy of the result requires special investigation in each case. Additions and subtractions are simple. If, for instance, the values of a and b, correct to two places of decimals, are 3.58 and 1.34, then 2.24, as the value of a—b, is not necessarily correct to two places. The limit of error of each being •005, the limit of error of their sum or difference is .or. For multiplication we make use of the See also:

• FORMULA
• FORMULA (Lat. diminutive of forma, shape, pattern, &c., especially used of rules of judicial procedure)

At the close of the work the extra figures dropped, the last figure which remains being corrected (§ 82 if necessary. 84. Roots and Surds.—The pth root of a number 043) may, if the number is an integer, be found by expressing it in terms of its prime factors; or, if it isnot an integer, by expressing it as a fraction in its lowest terms, and finding the pth roots of the numerator and of the denominator separately. Thus to find the cube root of 1728, we write it in the form 26.33, and find that its cube root is 22.3 = 12; Cr, to find the cube root of 1.728, we write it as iii=1--= 235 3' and find that the cube root is 253 =1.2. Similarly the cube root of 2197 is 13. But we cannot find any number whose cube is 2000. It is, however, possible to find a number whose cube shall approximate as closely as we please to 2000. Thus the cubes of 12.5 and of 12.6 are respectively 1953'125 and 2000.376, so that the number whose cube differs as little as possible from 2000 is somewhere between 12.5 and 12.6. Again the cube of 12.59 is 1995.616979, so that the number lies between 12.59 and 12.6o. We may therefore consider that there is some number x whose cube is 2000, and we can find this number to any degree of accuracy that we please. A number of this kind is called a surd; the surd which is the pth root of N is written jN, but if the index is 2 it is usually omitted, so that the square root of N is written %IN. 85.

Surd as a Power.—We have seen (§§ 4344) that, if we take the successive powers of a number N, commencing with 1, they may be written N°, N1, N2, N3, . . . , the series of indices being the standard series; and we have also seen (§ 44) that multiplication of any two of these numbers corresponds to addition of their indices. Hence we may insert in the power-series numbers with fractional indices, provided that the multiplication of these numbers follows the same law. The number denoted by Ni will therefore be such that N1XNAXN0=NA+;+3=N; i.e. it will be the cube root of N. By analogy with the notation of fractional numbers, Ni will be NI+i = Ni X Ni; and, generally, NQ will mean the product of p numbers, the product of q of which is equal to N. Thus Na will not mean the same as Ni, but will mean the square of Ni; but this will be equal to N3, i.e. (;l N)2= i/N. 86. Multiplication and Division of Surds.—To add or subtract fractional numbers, we must reduce them to a common denominator ; and similarly, to multiply or divide surds, we must express them as power-numbers with the same index. Thus ,72 X 5 = 2AX52 =26X56=48X125i=5006=64 500. 87.

Antilogarithms.—If we take a fixed number, e.g. 2, as base, and take as indices the successive decimal numbers to any particular number of places of decimals, we get a series of antilogarithms of the indices to this base. Thus, if we go to two places of decimals, we have as the integral series the numbers 1, 2, 4, 8, . . . which are the values of 2°, 21, 22, . . . and we insert within this series the successive powers of x, where x is such that x'0° = 2. We thus get the numbers 2.01, 2.02, 2.03, . . . , which are the See also:

• ANTI, or CAMPA

In the same way, by dividing by powers of 10 we may get negative indices. 88. Logarithms.—If N is the antilogarithm of p to the base a, i.e. if N= a'', then p is called the logarithm of N to the base a, and is written loge. N. As the table of antilogarithms is formed by successive multiplications, so the logarithm of any given are (i) ) number is in theory found by successive divisions. Thus, to find the logarithm of a number to base 2, the number being greater than 1, we first divide repeatedly by 2 until we get a number between r and 2; then divide repeatedly by 10,12 until we get a number between 1 and 10^12; then divide repeatedly by 1oo312; and so on. If, for instance, we find that the number is approximately equal to 23 X ('°s,12) 5 X ('°N2) 7 X (IOOO-^12) 4, it may be written 23.574 and its logarithm to base 2 is 3.574. For a further explanation of logarithms, and for an explanation of the treatment of cases in which an antilogarithm is less than I, see LOGARITHM. For practical purposes logarithms are usually calculated to base lo, so that loglo to=1, loglo too= 2, &c. IX. UNITS 8q. See also:

• CHANGE (derived through the Fr. from the Late Lat. cambium, cambiare, to barter; the ultimate derivation is probably from the root which appears in the Gr. «a urmu', to bend)

4d. as shillings and pence and also the expression of 3067S. 4d. as L s. and d. The usual statement is that to express £153, 7s. as shillings we multiply 153 by 20 and add 7. This, as already explained (§37), is incorrect. £153 denotes 153 units, each of which is £1 or 2os.; and therefore we must multiply 20S. by 153 and add 7s., i.e. multiply 20 by 153 (the unit being now Is.) and add 7. This is the expression of the process on the grouping method. On the counting method we have a scale with every 2oth shilling marked as a ; there are 153 of these 20's, and 7 over. The simplest case, in which the quantity can be expressed as an integral number of the largest units involved, has already been considered (§§ 37, 42). The same method can be applied in other cases by regarding a quantity expressed in several de- nominations as a fractional number of units of the largest denomination men- tioned; thus 7S. 4d. is to be taken as meaning 73,zs., but £o, 7S. 4d. as L zo (§ r 7). The reduction of £153, 7s.

4d. to pence, and of 368o8d. to L s. d., on this principle, is shown in diagrams A and B above. For reduction of pounds to shillings, or shillings to pounds, we must consider that we have a multiple-table (§ 36) in which the multiples of £I and of 20S. are arranged in parallel columns; and similarly for shillings and pence. 90. Change of Unit.—The statement " £153=3o6os." is not a statement of equality of the same kind as the statement " 153)00=3060," but only a statement of equivalence for certain purposes; in other words, it does not convey an absolute truth. It is therefore of interest to see whether we cannot replace it by an absolute truth. To do this, consider what the ordinary processes of multiplication and division mean in reference to concrete objects. If we want to give, to 5 boys, 4 apples each, we are said to multiply 4 apples by 5. We cannot multiply 4 apples by 5 boys, for then we should get 20 " boy-apples," an expression which has no meaning. Or, again, to distribute 20 apples amongst 5 boys, we are not regarded as dividing 20 apples by 5 boys, but as dividing 20 apples by the number 5. The multiplication or division here involves the omission of the unit " boy," and the operation is incomplete. The complete operation, in each case, is as follows. (i) In the case of multiplication we commence with the conception of the number " 5 " and the unit " boy "; and we then convert this unit into 4 apples, and thus obtain the result,20 apples.

The conversion of the unit may be represented as multiplication by a factor 4abo~es, so that the operation is 4apples X5boys= 5X41 boy SX 1 r boy=5X4apples =2oapples. Similarly, to convert £153 into shillings we must multiply it by a factor 2I, so that we get 20S. 20S. £1 X £153 =153 X £1 X £I = 153 X 20s. =3060S. Hence we can only regard £153 as being equal to 3o6os. if we regard this converting factor as unity. (ii) In the case of partition we can express the complete operation if we extend the meaning of division so as to enable us to divide 20 apples by 5 boys. We thus get 25 do s s = 4 i boy apples' which means that the See also:

• DISTRIBUTION (Lat. distribuere, to deal out)

See also:

• CORRESPONDENCE
• CORRESPONDENCE (from med. scholastic Lat. correspondentia, corrcspondere, compounded of Lat. cum, with, and respond ere, to answer; cf. Fr. correspondance)

, or the numbers 31,/1,3s/ 2, 311/3, ... or loglo poor, loglo I.0o2, loglo" 1.003 . . . we obtain a series in which each term corresponds with a term of the original number-series. This correspondence is usually shown by tabulation, i.e. by the formation of a table in which the original series is shown in one A B C column, and each term of the second series is placed in a second column op- posite the corresponding term of the first series, o •000 each column being headed r 1 •000 by a description of its 2 1.414 contents. It is sometimes 3 1.732 convenient to begin the first series with o, and even to give the series of nega- tive numbers; in most cases, however, these latter are regarded as belonging to a different series, and they need not be considered here. The diagrams, A, B, C are simple forms of tables; A giving a sum-series, B a multiple-series, and C a series of square roots, calculated approximately. 92. Correspondence of Numerical Quantities.—Again, in § 89, we have considered cases of multiple-tables of numerical quantities, where each quantity in one series is equivalent to the corresponding quantity in the other series. We might extend this principle to cases in which the terms of two series, whether of numbers or A I s. 12d. L1 20S. 368o8d. 3067S.

4d.